Subscribe to this blog

Follow by Email

Search This Blog

Handshakes and Pigeons

Q: A party has a hundred guests. The guests mingle freely and randomly shake hands with one another (no repeat handshakes). What is the probability that any two persons will have the same number of handshakes?Discrete Mathematics with Applications

A: In order to solve this, a powerful yet simple tool available is the "Pigeonhole Principle". At its heart its a rather simple statement. If you place \(n\) pigeons in \(m\) holes where \(n \ge m\) at least one hole must have more than one pigeon. This is simple to grasp if you look at the picture below which shows the case with 3 pigeons in 2 boxes.

A guest at the party can shake hands 0 to \(n - 1\) times. Each of these counts can be thought of as states a guest can get into and the resulting distribution is quite arbitrary. For example, there could be guests who shake hands with none, with 3 other guests and so on. Notice how this fits nicely with the pigeon hole principle. In such a setting, a guest can fit into any of the \(n - 1\) holes or states. Also note, we could get misled to think that there are a total of \(n\) states given that \([1..n-1]\) implies \(n-1\) states and the zero state is one more, leading to a total \(n\) states. However, if a guest shakes hands with none, no other guest can go beyond \(n-2\) handshakes. So the total can never be beyond \(n-1\) handshakes. This leads us to the conclusion that at least two guests will have the same number of handshakes. Thus the sought probability is 1.

If you are looking to buy some books in probability here are some of the best books to learn the art of Probability

Discovering Statistics Using R
This is a good book if you are new to statistics & probability while simultaneously getting started with a programming language. The book supports R and is written in a casual humorous way making it an easy read. Great for beginners. Some of the data on the companion website could be missing.

A Course in Probability Theory, Third Edition
Covered in this book are the central limit theorem and other graduate topics in probability. You will need to brush up on some mathematics before you dive in but most of that can be done online

Discovering Statistics Using R
This is a good book if you are new to statistics & probability while simultaneously getting started with a programming language. The book supports R and is written in a casual humorous way making it an easy read. Great for beginners. Some of the data on the companion website could be missing.

Linear Algebra (Dover Books on Mathematics)
An excellent book to own if you are looking to get into, or want to understand linear algebra. Please keep in mind that you need to have some basic mathematical background before you can use this book.

Linear Algebra Done Right (Undergraduate Texts in Mathematics)
A great book that exposes the method of proof as it used in Linear Algebra. This book is not for the beginner though. You do need some prior knowledge of the basics at least. It would be a good add-on to an existing course you are doing in Linear Algebra.

Follow @ProbabilityPuzIf you are looking to learn time series analysis, the following are some of the best books in time series analysis.

Introductory Time Series with R (Use R!)
This is good book to get one started on time series. A nice aspect of this book is that it has examples in R and some of the data is part of standard R packages which makes good introductory material for learning the R language too. That said this is not exactly a graduate level book, and some of the data links in the book may not be valid.

Econometrics
A great book if you are in an economics stream or want to get into it. The nice thing in the book is it tries to bring out a oneness in all the methods used. Econ majors need to be up-to speed on the grounding mathematics for time series analysis to use this book. Outside of those prerequisites, this is one of the best books on econometrics and time series analysis.